In chemical graph theory, a topological index is a value which characterizes a graph representing a molecule. This value is correlated with some properties of said molecule. A widely studied class of such indices is degree-based invariants. A well-known example of this is the Randic index. Many others were introduced subsequently. In this presentation, we present a general technique which potentially allows us to obtain bounds on such chemical graph invariants.

The arithmetic-geometric index is a newly proposed degree-based graph invariant in mathematical chemistry. We give a sharp upper bound on the value of this invariant for connected chemical graphs of given order and size and characterize the connected chemical graphs that reach the bound. We also prove that the removal of the constraint that extremal chemical graphs must be connected does not allow to increase the upper bound.

A graph G=(V,E) is antimagic if there is a bijection f:E→{1,2,…,|E|} such that all vertex sums are pairwise distinct, where the vertex sum is the sum of labels of all edges incident to a given vertex. It has been proven that regular graphs are antimagic, and more recently that a subclass of bipartite biregular graphs is antimagic as well. We show a new labelling algorithm that allows to extend the result to (k,2)-bipartite biregular graphs, as well as connected (k,l)-bipartite biregular graphs.